Why is modular exponentiation important in cryptography?

Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys. Modular exponentiation is efficient to compute, even for very large integers.

Is modular exponentiation one way?

Another well-known possibly-one-way function is modular exponentiation, whose inverse, the discrete logarithm, computes x such that bx≡y(modn), given y. The fact that xx, as a function from Z+ to Z+, is monotonically increasing makes finding x from xx relatively easy.

How do you do fast modular exponentiation?

How can we calculate A^B mod C quickly for any B?

Step 1: Divide B into powers of 2 by writing it in binary.
Step 2: Calculate mod C of the powers of two ≤ B.
Step 3: Use modular multiplication properties to combine the calculated mod C values.

What does MOD mean in math?

modulus
Given two positive numbers a and n, a modulo n (abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. The modulo operation is to be distinguished from the symbol mod, which refers to the modulus (or divisor) one is operating from.

How do you do modular arithmetic?

Modulus on a Standard Calculator

Divide a by n.
Subtract the whole part of the resulting quantity.
Multiply by n to obtain the modulus.

What is the time complexity of exponentiation?

Algorithms List of Mathematical Algorithms. Binary exponentiation is an algorithm to find the power of any number N raise to an number M (N^M) in logarithmic time O(log M). The normal approach takes O(M) time provided multiplication takes constant time.

Is squaring exponential?

Exponents and have special names. Raising a base to a power of is called “squaring” a number. Raising a base to a power of is called “cubing” a number. The inverse of squaring a number is finding the square root of a number.

What is modular arithmetic in cryptography?

Modular arithmetic is a system of arithmetic for integers, where values reset to zero and begin to increase again, after reaching a certain predefined value, called the modulus (modulo). Modular arithmetic is widely used in computer science and cryptography.

What is modular exponentiation used for In cryptography?

It is useful in computer science, especially in the field of public-key cryptography . The operation of modular exponentiation calculates the remainder when an integer b (the base) raised to the e th power (the exponent), be, is divided by a positive integer m (the modulus).

What is modular exponentiation in C programming?

The operation of modular exponentiation calculates the remainder when an integer b (the base) raised to the eth power (the exponent), be, is divided by a positive integer m (the modulus). In symbols, given base b, exponent e, and modulus m, the modular exponentiation c is: c ≡ be (mod m).

What is RSA cryptography?

Cryptography studies ways to share secrets securely, so that even eavesdroppers can’t extract any information from what they hear or network traffic they intercept. One of the most popular cryptographic algorithms called RSA is based on unique integer factorization, Chinese Remainder Theorem and fast modular exponentiation.

How do you calculate modular exponents?

The most straightforward method of calculating a modular exponent is to calculate be directly, then to take this number modulo m. Consider trying to compute c, given b = 4, e = 13, and m = 497: One could use a calculator to compute 4 13; this comes out to 67,108,864.